11 problems

Let $S_n$ be the set of monic complex polynomials of degree $n\ge 2$ whose zeros lie in the closed unit disk, and let … A polynomial is locally maximal for Sendov's conjecture if i…

Let $\beta$ be a complex number with $|\beta|\le1$, and let $S(\beta)$ denote the set of polynomials of degree at least $2$ with complex coefficients, all roots in the closed unit…

Let $\beta\in[0,1]$, let $S(\beta)$ be the set of polynomials of degree at least $2$ with complex coefficients, all roots in the closed unit disk, and at least one root at $\beta$,…

Let $\beta\in[0,1]$, let $S(\beta)$ be the set of polynomials of degree at least $2$ with complex coefficients, all roots in the closed unit disk, and at least one root at $\beta$,…

Let $\beta\in[0,1]$, let $S(\beta)$ be the set of polynomials of degree at least $2$ with complex coefficients, all roots in the closed unit disk, and at least one root at $\beta$,…

For $n\ge2$, let $r_n(\beta)$ be the maximum of $d(P,\beta)$ over $P\in S(\beta)$ of degree $n$, and let $c_n$ denote the corresponding sharp quadratic-refinement coefficient in th…

Let $\beta$ be a real number in $[0,1]$. For a polynomial $P$ of degree at least $2$ with complex coefficients, all roots in the closed unit disk, and at least one root at $\beta$,…

Let $f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0\in\mathbb{C}[x]$ have no repeated zeros, and let $\mathcal{S}=(f(x),f(x),\ldots,f(x))$. Let…

Minimal-support variance conjecture.

Schmeisser's conjecture. For every $\zeta\in\operatorname{conv}(V(F))$, the closed disk centered at $\zeta$ with radius $\sigma_\infty(F)$ contains a critical point of $F$.

Borcea's variance conjecture. For every such $F$ and every $p\in[1,\infty)$,